Impedance rule



Feb. 3, 1953 N. E. SPRECHER 2,627,114

IMPEDANCE RULE Filed Feb. :5. 1949 3mm Re 1 N04 5. SPREOHER FIG. 3 I

Patented Feb. 3, 1953 UNITED STATES PATENT OFFICE 3'Claims. (Cl. 33-76) (Granted under Title '35, U. S. Code (1952),

sec. 266) 1 The inventicn described herein ma be mamfact'ured and used by or for the Government 'for governmental purposes without the payment of any royalty thereon. n

The present invention relates to an impedance rule and more "particularly to a rule for solving impedance problems for electrical circuits and networks.

An object of the prevent invention is the provision of a rule for rapidly determining the values of circuit elements in both series and parallel circuits.

Another object is to provide a rule for calculating the amplitude and phase angle of the impedance of either series or parallel networks.

A further object is to provide a rule for determining the values of the circuit constants necessary for transforming a resistance from one value to another. I

Other objects and many of the attendant advantages of this invention will be readily appreciated as the same become better understood by reference to the following detailed description when considered in connection with the accompanying drawings in which like reference chara'cters designate like parts throughout the figures thereof and wherein:

Fig. 1 is a plan view illustrating a preferred embodiment of the impedance rule of the subject Fig. 3 shows a diagrammatic view of the im-v pedance rule illustrating its operation for resistance transformation.

Referring now to the drawings there is shown in Fig. 1 a supporting member ll made of any l'suitable durable material and having inscribed or printed thereon a chart I2 of rectilinear coordinates. The chart I2 is preferably square in shape and comprises 100 squares each of which is divided into 100 smaller squares.

Drawn on chart l2 in any suitable manner is an are 13 of a circle having its center at one corner 14 of the chart and its radius equal to the length of one side of chart l2, the are [3 being divided into ninety degrees as indicated in Fig. 1. Also drawn on chart I2 is a semicircle I5 having a radius equal to one half the length of one side of chart I2 and its center positieiied at the midpoint It of said side.

Pivotally mounted on member I l at corner [4 is a straightedge ll preferably made of transparent plastic material and having inscribed theren a series biha'ii lines I8 spaced apart the same length as the scale on chart l2, and a hairline 22 runhing along the longitudinal centerline thereof. Straightedg'e I1 is mounted in a bearing !9, preferably made of metal, by means of pin 2! passing through straightedge ll, bearing l8 and memher I l as clearly shown in Fig. 2.

Slidably mounted on straightedge H is a straightedge 23, preferablyof transparent plastic material, having a hairline 24 inscribed along the longitudinal centerlin'e thereof. Straightedge 23 is mounted on edge I! by means of flanges 25, hairline 24 being always aligned at right angles to the longitudinal centerline of straighted'ge H.

The rule of the present invention may be used to solve numerous impedance problems in electrical circuits and networks, several of which will be illustrated be1o'w.

In solving a series impedance problem, the magnitude of the impedance of a series circuit consisting of resistance RS and reactance X5 is al V a The phase angle of the impedance may be designated as I Thus, to determine these values on chart l2, the vertical line representing the resistance and the horizontal line representing the reactance are located, and straightedge I! is swung on its pivot point so that hairline 22 coincides with the intersection of these two lines, the amplitude of the series impedance being then read on straightedge IT. The intersection of hairline 22 with arc I3 on chart I2 indicates the phase angle of the impedance. 7

In solvingaparallel impedance problem, the magnitude of the impedance of a parallel circuit Tan 0,

consisting of resistance Rp and reactance Xe is r 'il=-:- p +X Tan 65:

Therefore, to determine the impedance of a resistance and reactance in parallel, slide straightedge 23 along straightedge I! and at the same time adjust the angle of straightedge II until hairline 24 intersects the lower edge of chart I2 at a point equal to the resistance Rp and at the same time intersects the left edge of chart :2 at a point equal to the reactance Xp. The point of intersection of hairlines 22 and 24, as measured on straightedge I1, is the altitude of the right triangle and is equal to the magnitude of the impedance Zp, while the intersection of hairline 22 and are I3 indicates the phase angle.

If it is desired to determine the equivalent series circuit for the parallel circuit above, read down vertically from the parallel impedance point to determine the equivalent series resistance and read horizontally to the left edge of chart I2 to obtain the equivalent series reactance. Similarly, if it is desired to determine the equivalent parallel impedance for the series circuit of the first problem, straightedge 23 is moved along straightedge II until hairline 22 and 2 intersect, and the intersections of hairline 24 with the lower and left edges of chart I2 determine the equivalent parallel resistance and reactance, respectively.

By utilizing semicircle I5, the rule may be efiiciently used to transform resistances from one effective value to another, this problem being solved on a ratio basis. Thus, the larger resistance is multiplied by some factor so that it may be represented by 10 units on the rule, the smaller resistance being multiplied by this same factor and the resistance values obtained being used on chart I2. The factor by which the original values were altered must be used inversely on all values obtained from the rule to present the final answer.

If it is desired to transform a given resistance to a higher effective resistance, this can be accomplished by connecting an inductive reactance X1. in series with the resistance R and connecting a capacitive reactance X in parallel with the series circuit of R and XL. The impedance of the parallel circuit is By multiplying the terms out and clearing the denominator of its imaginary term, the value of the impedance is X, RI-JX,(X,X1,R Xz, (I) Ram-X.)

In order that the impedance Z be equal to'the effective resistance desired Re, the phase angle 0 must be zero or Therefore,

X 2+ 2 X6 XL Substituting this value of Xe in Equation 1 gives 2 2 Z=R,= :23 sec 0:

where ZR is the amplitude of the impedance of R and XL in series and a is the phase angle of said impedance.

Similarly, from Equation 2 Xc ZR csc. (1.

Referring now to Fig. 3, it is seen that in order for Re to remain constant at a value ID, as WGQ above, Re must equal the value of the diameter of semicircle I5 and the intersection on semicircle I5 of a vertical line from the value of resistance R determines the value of the inductance X1. to be added in series with resistance R. The intersection of hairline 24 with the left edge of chart I2 determines the value of the capacitive reactance Xe, as seen in Fig. 3.

To illustrate this operation of the rule, let us assume that it is desired to transform a 2 ohm resistance to a 20 ohm resistance. Dividing 20 and 2 by a factor of 2 results in Re equal to 10 and R equal to 1. Now lay the hairline 24 of straightedge 23 from 10 on the bottom scale, as shown in Fig. 3, through the intersection of semicircle I5 and the vertical line representing 1 ohm resistance. Reading horizontally from this intersection, the value of the inductive reactance is 3 ohms, and the intersection of hairline 24 with the left edge of chart I2 gives a value of 3 ohms capacitive reactance. Multiplying these values by a factor of 2 gives the final values of 6 ohms inductive reactance in series and 6% ohms capacitive reactance in parallel with the series circuit.

The rule may also be used to transform a high resistance to a lower effective resistance. This may be accomplished by connecting an inductive reactance X1. across the resistance and connecting a capacitive reactance Xe in series with the parallel circuit. Calculations of the equivalent impedance of this circuit results in the following equations:

and

Xc=lZpl SEC a where |Z equals the magnitude of the impedance of the parallel circuit of resistance R and inductive reactance XL, and a. is the phase angle of this impedance.

Therefore, to determine these values on the rule, the rule is set up as in the previous problem with the resistances set up in opposite order, that is the known resistance R along the horizontal axis and the desired equivalent resistance Re along the vertical axis. The intersection of hairline 24 and the left edge of chart I2 determines the inductive reactance XL, and the horizontal reading from the point of intersection of hairline 24 and semicircle I5 determines the value of the capacitive reactance.

The sine, cosine and tangent of any angle can also be obtained simply by use of this rule. By locating the desired angle on the arc I3 of chart I2 and then taking the horizontal and vertical readings from this point, the sine and cosine, respectively, are obtained. The tangent is found by swinging straightedge I I to the desired angle and reading the intersection of hairline 22 with the right edge of chart I2. If the reading is off the chart, a ratio of vertical to horizontal readings may be utilized, or the next smaller division is read and the value obtained is multiplied by the proper factor.

Various modifications are contemplated and may obviously be resorted to by those skilled in the art without departing from the spirit and scope of the invention, as hereinafter defined by the appended claims, as only a preferred embodiment thereof has been disclosed.

What is claimed is:

1. An impedance rule comprising a chart of rectangular coordinates, an arc on said chart having its center at one corner thereof and its radius equal to the length of One side thereof, a

semicircle on said chart having its center at the midpoint of the lower horizontal edge thereof and its radius equal to one half the length of said side, said semicircle intersecting the center of said are, a first straightedge pivotally mounted on said chart at said corner, and a second straightedge slidably mounted on said first straightedge for movement at right angles thereto, and in operative relation to said semicircle.

2. An impedance rule according to claim 1, wherein said first straight edge has a scale inscribed thereon, the divisions of said scale being equal to the divisions of said chart, and wherein said are is divided into degrees.

3. An impedance rule comprising a substantially flat member, a chart of horizontal and vertical rectilinear coordinates inscribed thereon, an arc inscribed on said member and having its center at the intersection of one of said horizontal and one of said vertical coordinates, said are intersecting each of said last mentioned horizontal and vertical coordinates, a semicircle inscribed on said member having its midpoint along one of said last mentioned coordinates, said semicircle intersecting said center of said arc, a straightedge pivotally mounted on said member at the center of said arc, and a second straight- 6 edge slidably mounted on the first-mentioned straightedge in operative relation to said semicircle and for movement at right angles with respect to said first-mentioned straightedge.

NOAH E. SPRECHER.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS 

